2013/11/30 by Ulrich Bauer, Michael Lesnick · 2 citations
Mathematics · Computer Science · #math.AT #cs.CG #math.AC #msc:13P20 #msc:55U99
paper · pdf · doi:10.20382/jocg.v6i2a9
published as Journal of Computational Geometry 6:2 (2015), 162-191 · Expanded journal version, to appear in Journal of Computational Geometry. Includes a proof that no definition of induced matching can be fully functorial (Proposition 5.10), and an extension of our single-morphism characterization of the interleaving relation to multidimensional persistence modules (Remark 6.7). Exposition is improved throughout. 11 Figures added
arxiv created 2016/04/30 · arxiv updated 2016/10/25
We define a simple, explicit map sending a morphism f:M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and \mathopcoker f. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel "single-morphism" characterization of the interleaving relation on persistence modules.